Acta Math. Univ. Comenianae
Vol. LXIV, 1(1995), pp. 123–139
123
ERGODIC AVERAGES AND INTEGRALS OF COCYCLES
H. B. KEYNES, N. G. MARKLEY and M. SEARS
Abstract. This paper concerns the structure of the space C of real valued cocycles for a flow (X, Zm ). We show that C is always larger than the set of cocycles
cohomologous to the linear maps if the flow has a free dense orbit. By considering appropriate dual spaces for C, we obtain the concept of an invariant cocycle
integral. The extreme points of the set of invariant cocycle integrals parallel the
role of ergodic measures and enable us to investigate different ergodic averages for
cocycles and the uniform convergence of such averages. The cocycle integrals also
enable us to characterize the subspace of the closure of the coboundaries in C, and
to show that C is the direct sum of this space with the linear maps exactly when
the invariant cocycle integral is unique.
1. The Space of Cocycles
Let X be a compact metric space and let Zm denote the integer lattice in Rm ,
m-dimensional Euclidean space. We will assume that Zm acts as a group of homeomorphisms on X, that is we have a flow (X, Zm ). A real valued (topological)
cocycle for such a flow is a continuous function h : X × Zm → R such that for all
x ∈ X and all a, b ∈ Zm
h(x, a + b) = h(x, a) + h(ax, b)
where ax denotes the action of a on x. This equation is called the cocycle equation. The theme of this paper is to investigate ergodic averages of these cocycles
and to describe an appropriate setting for invariant integrals of cocycles.
In earlier papers ([2], [4], and [5]), we have studied vector valued cocycles and
their role in understanding the structure of Rm actions on compact metric spaces.
Since the coordinates of a vector valued cocycle are real valued cocycles, the results
in this paper provide additional tools for the analysis of Rm actions. Specifically
they provide a natural definition of a nonsingular Rm valued cocycle. K. Madden
and the second author [7] have shown that the suspension flow of a nonsingular
cocycle is a time change of the constant one suspension and that every Rm action
with a free dense orbit has an almost one-to-one extension which is the suspension
Received March 3, 1994; revised February 27, 1995.
1980 Mathematics Subject Classification (1991 Revision). Primary 58F25; Secondary 28D10,
54H20.
124
H. B. KEYNES, N. G. MARKLEY and M. SEARS
of a nonsingular cocycle. However, the present work stands on its own as an
extension of classical results about ergodic averages and coboundaries for a single
homeomorphism.
Let C denote the set of real valued cocycles for (X, Zm ). Clearly C is a vector
P
m
space over R. Letting |a| = m
i=1 |ai | for a = (a1 , · · · , am ) ∈ Z ,
khk = sup
|h(x, a)|
:x∈X
|a|
and a ∈ Zm
defines a norm. Let e1 , · · · , em denote the usual generators of Zm . Using the
cocycle equation it is not hard to show that
khk = sup{|h(x, ej )| : x ∈ X
and 1 ≤ j ≤ m},
and that with this norm C is a separable Banach space.
Remark 1.1. If h ∈ C, then
|h(x, a) − h(x, b)| ≤ khk |a − b|
for all x ∈ X and a, b ∈ Zm .
Proof. Clearly |h(y, c)| ≤ khk |c| for any y and c. Since h(x, b) + h(bx, a − b) =
h(x, a), we have
|h(x, a) − h(x, b)| = |h(bx, a − b)| ≤ khk |a − b|.
If h ∈ C and h(x, a) = h(y, a) for all x, y ∈ X and a ∈ Zm , then the map
a → h(x, a) is linear. Conversely, if T ∈ L, the linear maps from Rm into R, then
h(x, a) = T (a) defines a cocycle in C. Consequently, L is a closed subspace of C
which is called the constant cocycles. In particular, the dual basis e∗1 , · · · , e∗m of
e1 , · · · , em are constant cocycles. We now seemingly have two norms on L; namely
the cocycle norm,
kT k = sup{|T (ej )| : 1 ≤ j ≤ m}
and the linear functional norm,
|T | = sup{|T (v)| : |v| ≤ 1}
P
where as above |v| = m
i=1 |vi |. However, it is easily verified that in our context
they are equal, and we will use the notation kT k.
Let C(X) denote the Banach space of continuous real valued functions on X
with the usual norm. Given f ∈ C(X) we can construct h ∈ C by setting
h(x, a) = f (ax) − f (x).
ERGODIC AVERAGES AND INTEGRALS OF COCYCLES
125
Such a cocycle is called a coboundary and the coboundaries, B, form another
subspace of C. It is important because two cocycles that differ by a coboundary
are in many ways indistinguishable.
When m = 1 we have a single homeomorphism ϕ of X generating the Z action.
In this case C is isometrically isomorphic to C(X) because given f ∈ C(X)
 Pn−1
k

 k=0 f (ϕ (x))
h(x, n) = 0

 P−n
− k=1 f (ϕ−k (x))
n>0
n=0
n<0
defines a cocycle with h(x, 1) = f (x). Notice that
n−1
h(x, n)
1X
=
f (ϕk (x))
n
n
k=0
is an ergodic average for f (x) = h(x, 1). Moreover, these averages converge uniformly in x for all f if and only if (X, ϕ) is uniquely ergodic. (See §6.5 in Walters
[9] for a general discussion of unique ergodicity.)
In general we can define a Zm action on C by setting ah(x, b) = h(ax, b) =
h(x, a + b) − h(x, a) and then
1
Nm
X
ah(x, b)
0≤ai <N
is a kind of an ergodic cocycle average and one expects its limits to be related
to integration. But it is not clear how to interpret h(x, a)/|a| as an ergodic average and fit it into an integration theory for cocycles. The main results in this
paper establish the concept of a cocycle integral which firmly links these two types
of averaging together for cocycles in the topological setting. Furthermore, these
integrals provide a characterization of B, and the equivalence of uniform convergence and unique cocycle integral holds similarly to the equivalence of uniform
convergence and unique invariant measure.
2. Existence of Non-Trivial Cocycles
A constant cocycle plus a coboundary is neither a constant cocycle nor a
coboundary. Are there cocycles which are not of this form, that is, are there
cocycles which are not cohomologous to a constant cocycle? In this section we will
prove the unpublished folklore theorem that these cocycles are usually dense in C
because without it the rest of our results are pointless.
The linear map S : C(X) → C given by (Sf )(x, a) = f (ax) − f (x) is bounded
and has range B. The kernel of S consists of the continuous functions which are
126
H. B. KEYNES, N. G. MARKLEY and M. SEARS
constant on orbits and hence constant on orbit closures. Because L is a closed
subspace of C, we can form the quotient Banach space C˜ = C/L and let π be
˜ Note that C = B + L (every cocycle
the bounded linear projection of C onto C.
is cohomologous to a constant cocycle) if and only if π ◦ S is onto. Hence, the
following theorem will answer the question:
Theorem 2.1. If (X, Zm ) has a free dense orbit, then π ◦ S : C(X) → C˜ is not
onto.
Proof. Since (X, Zm ) has a dense orbit, the kernel K of S is the constant
˜ If π ◦ S
functions and there is an induced bounded linear map S̃ : C(X)/K → C.
is onto, then S̃ is an isomorphism by the open mapping theorem. In particular,
S̃ −1 is continuous and if gn is a sequence in C˜ such that kgn k converges to 0, then
kS̃ −1 gn k also converges to 0. The following lemma establishes the existence of a
sequence which violates this condition and completes the proof of the theorem by
contradiction. (This proof is an adaption of a proof for m = 1 which R. Zimmer
showed us.)
Lemma 2.2. If (X, Zm ) has a free orbit, then given > 0 there exists f ∈ C(X)
such that kS(f )k ≤ and inf{kf + ck : c ∈ R} = 1/2.
Proof. Suppose 0 < < 1 and the orbit of x0 is free, so a → ax0 is one-toone. Choose M , a positive integer, such that (1 − )M < . There exists an open
neighborhood U of x0 such that aU ∩ bU = φ when −(M + 1) ≤ ai , bi ≤ M + 1
and there exists f0 ∈ C(X) such that f0 (x0 ) = 1, f0 (x) = 0 for x 6∈ U , and
0 ≤ f0 (x) ≤ 1 for all x. Define fa by fa (x) = (1 − )|a| f0 ((−a)x) and set
f (x) =
X
{fa (x) : |ai | ≤ M, i = 1, · · · , m}.
Note that f (x) = 0 unless x ∈ aU for some a with |ai | ≤ M , i = 1, · · · , m, in
which case f (x) = fa (x).
Next we show that kSf k = sup{|f (ei x) − f (x)| : x ∈ X and i = 1, · · · , m} ≤ .
There are several cases for each ei . First suppose f (ei x) and f (x) are not zero.
Hence x ∈ aU , ei x ∈ (a+ei)U , |aj | ≤ M for j 6= i, and −M ≤ ai < ai +1 ≤ M +1.
When 0 ≤ ai we have
f (ei x) − f (x) = (1 − )|a|+1 f0 ((−a)x) − (1 − )|a| f0 ((−a)x)
= −(1 − )|a| f0 ((−a)x)
and
|f (ei x) − f (x)| ≤ .
A similar calculation holds for ai < 0.
ERGODIC AVERAGES AND INTEGRALS OF COCYCLES
127
Now suppose f (x) 6= 0 and f (ei x) = 0. In this case x ∈ aU with |aj | ≤ M for
j 6= i and ai = M . Thus |a| ≥ M and
|f (ei x) − f (x)| = | − (1 − )|a| f0 ((−a)x)| ≤ (1 − )M < .
A similar calculation works when f (ei x) 6= 0 and f (x) = 0.
Finally we calculate inf{kf + ck : c ∈ R} = 1/2. First note that kf + ck =
sup{|f + c| : x ∈ X} ≥ 1 for c ≥ 0 and c ≤ −1. For −1 < c < 0 it is the maximum
of −c and 1 + c which takes its minimum value when c = −1/2.
Theorem 2.3. Every open set in C contains cocycles which are not cohomologous to a constant cocycle.
˜ If π(V ) ⊆
Proof. Let V be an open set in C. Then π(V ) is open in C.
π ◦ S(C(X)), then π ◦ S would be onto.
The following result also follows readily from the above lemma.
Theorem 2.4. If (X, Zm ) has a free dense orbit, then B is not closed.
Proof. If B was closed, than the induced map of C(X)/K onto B would be a
Banach isomorphism. Lemma 2.2 shows that the inverse would not be continuous.
Thus B is not closed.
In contrast to Theorem 2.3, there is the question of what kind of functions are
in L + B. For example, when the underlying system is an irrational rotation of
the circle, then cos nθ and sin nθ are in B [3, p. 563] and consequently, all finite
trigonometric polynomials are in L + B. In a recent paper Schmidt [8] has shown
that for some subshifts of finite type cocycles with a summable variation are in
L + B. He also gives some concrete examples of cocycles not in L + B.
3. Invariant Linear Functionals on C
There is a natural map from C into C(X)m which we can use to study C ∗ ,
the dual of C. Define Θ : C → C(X)m by Θ(h) = (h(·, e1 ), · · · , h(·, em )) where
h(·, ej ) denotes the function x → h(x, ej ). Using the norm k(f1 , · · · , fm )k =
sup kfj k, Θ is an isometric isomorphism of C onto the closed subspace
1≤j≤m
{(f1 , · · · , fm ) : fi (ej x) − fi (x) = fj (ei x) − fj (x)
for i 6= j}
of C(X)m .
The dual of C(X)m can be identified with C ∗ (X)m . Explicitly, given
f = (f1 , · · · , fm ) ∈ C(X)m
and γ = (γ1 , · · · , γm ) ∈ C ∗ (X)m ,
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H. B. KEYNES, N. G. MARKLEY and M. SEARS
set
γ(f ) =
m
X
γi (fi ).
i=1
Moreover,
kγk = sup{|γ(f )| : kfk = sup kfi k ≤ 1} =
1≤j≤m
m
X
kγi k.
i=1
Proposition 3.1. If γ ∈ C ∗ , then there exists γ = (γ1 , · · · , γm ) ∈ C ∗ (X)m
with the following properties
a)
b)
c)
d)
γ(h) = γ(Θ(h))
kγk = kγk
γ(e∗j ) = γj (1)
|γ(e∗j )| ≤ kγj k.
Proof. Use the Hahn-Banach theorem for a) and b). For c) note that Θ(e∗j ) =
(0, · · · , 0, 1, 0, · · · , 0), and then d) follows from c).
P
m
Theorem 3.2. If γ ∈ C ∗ , then kγ | Lk = i=1 |γ(e∗i )| and the following are
equivalent
a) kγ | Lk = kγk
Pm
b) kγk = j=1 |γ(e∗j )|
c) there exist Borel probability measures µ1 , · · · , µm such that
γ(h) =
m
X
j=1
γ(e∗j )
Z
X
h(x, ej ) dµj .
Proof. First we calculate kγ | Lk. Let T ∈ L with kT k = sup |T (ej )| ≤ 1
1≤j≤m
P
∗
and write T = m
T
(e
)e
.
Then
i i
i=1
|γ(T )| = |
m
X
i=1
T (ei )γ(e∗i )| ≤
m
X
|γ(e∗i )|
i=1
Pm
and kγ | Lk ≤ i=1 |γ(e∗i )|. Now define T by setting T (ei ) = 1, −1, or 0 according
Pm
as γ(e∗i ) is positive, negative or zero. Then kT k = 1 and γ(T ) = i=1 |γ(e∗i )| ≤
P
∗
kγ | Lk. It follows that kγ | Lk = m
i )| and a) and b) are equivalent.
i=1 |γ(e
P
m
If c) holds, then it follows that kγk ≤ j=1 |γ(e∗j )| = kγ | Lk ≤ kγk and a)
holds. So it remains to show that b) implies c).
ERGODIC AVERAGES AND INTEGRALS OF COCYCLES
129
Assuming b), let γ = (γ1 , · · · , γm ) ∈ C ∗ (X)m be given by Proposition 3.1. We
now have
m
m
X
X
kγk = kγk =
kγj k ≥
|γ(e∗j )| = kγk
j=1
j=1
follows that |γj (1)| = |γ(e∗j )| = kγj k.
γ(e∗i ) 6= 0 consider (1/γ(e∗i ))γi = µi
and it
∈ C ∗ (X). Clearly kµi k = 1 = µi (1).
For
By the Riesz Representation
Theorem µi is given by a Borel probability measure,
R
that is, µi (f ) = X f (x) dµi . For γ(e∗j ) = 0 let µj be any Borel probability
measure. Finally
m
X
γ(h) = γ(Θ(h)) =
=
X
γ(e∗j )
γ(e∗
j )6=0
=
m
X
j=1
γj (h(·, ej ))
j=1
γ(e∗j )
1
γj (h(·, ej ))
γ(e∗j )
Z
X
h(x, ej ) dµj
to complete the proof.
R
For η ∈ C ∗ (X) it is clear that η(1) = kηk if and only if η(f ) R= X f (x) dµ for
some Borel measure µ. This can
R be rephrased as follows: η(f ) = X f (x) dµ for all
f ∈ C(X) or η(f ) = − X f (x) dµ for all f ∈ C(X) if and only if
kη | constant functionsk = kηk. Hence the elements γ ∈ C ∗ such that kγ | Lk =
kγk provide analogs of measures for C and can be viewed as integration in the
Pm
∗
i=1 γ(ei )ei direction. (An ordinary Borel measure providing integration in the
positive direction of R).
The natural action of Zm on C, given by ah(x, b) = h(ax, b), is clearly norm
preserving and hence continuous. We say γ ∈ C ∗ is invariant if γ(ah) = γ(h) for
all a ∈ Zm and h ∈ C.
Proposition 3.3. If γ ∈ C ∗ is invariant, then there exists γ = (γ1 , · · · , γm ) ∈
C (X)m such that
∗
a)
b)
c)
d)
e)
γ(h) = γ(Θ(h))
kγk = kγk
γ(e∗j ) = γj (1)
|γ(e∗j )| ≤ kγj k
γj is invariant.
Proof. Apply Proposition 3.1 to get γ 0 satisfying a) through d), and observe
that
X
1
γN = m
aγ 0
N
0≤ai <N
130
H. B. KEYNES, N. G. MARKLEY and M. SEARS
also satisfies a) through d) because γ is invariant. There exists a subsequence of
γ N converging coordinate-wise in the weak∗ topology and its limit will satisfy a)
through e).
Given w ∈ Rm with |w| = 1, we define the invariant cocycle integrals in the w
direction by
I(w) = {γ ∈ C ∗ : kγk = 1, γ(e∗i ) = wi , γ is invariant}.
Theorem 3.4. If γ ∈ I(w), then there exist invariant Borel probability measures µ1 , · · · , µm on X such that
Z
m
X
γ(h) =
wj
h(x, ej ) dµj .
j=1
X
Furthermore, I(w) is non-empty, convex, and compact in the weak∗ topology on C ∗ .
Proof. Let γ ∈ I(w). Clearly
m
X
|γ(e∗j )| =
j=1
m
X
|wj | = |w| = 1 = kγk
j=1
and Theorem 3.2 applies using the γ given by Proposition 3.3 to produce the
required invariant measures. The rest is obvious.
We now have a set of invariant cocycle integrals for each direction in Rm and it
is natural to ask if there is a unifying concept of cocycle integral from which the
I(w)0 s can be extracted. This requires a different dual of C.
4. Invariant Cocycle Integrals
In this section we define a new dual of C based on the idea that the integral of
a cocycle should be a constant cocycle. Define C # = {σ : C → L : σ is bounded
and linear} with norm
kσk = sup kσ(h)k = sup sup |σ(h)(w)|
khk≤1
khk≤1 |w|≤1
where h ∈ C and w ∈ Rm . Clearly C # is a Banach space.
We can also define a weak∗ topology on C # by σn → σ if and only if σn (h) →
σ(h) for all h ∈ C. As usual a neighborhood base at σ0 is all sets of the form
{σ : |σ(hi ) − σ0 (hi )| < , i = 1, · · · , k}.
The proof of Alaoglu’s Theorem applies here and the unit ball in C # is weak∗
Pm
compact. For x ∈ X we define σx ∈ C # by σx (h) = j=1 h(x, ej )e∗j . It follows
that kσx k = 1, σx (T ) = T for T ∈ L, and x → σx is a homeomorphism of X into
C # with the weak∗ topology.
ERGODIC AVERAGES AND INTEGRALS OF COCYCLES
131
Theorem 4.1. Let σ ∈ C # . If for all T ∈ L, σ(T ) = kσkT , then there exist
Borel probability measures µ1 , · · · , µm on X such that
σ(h) = kσk
m Z
X
j=1
X
h(x, ej ) dµj e∗j .
Proof. Set γi (h) = σ(h)(ei ). Clearly γi ∈ C ∗ and σ(h) =
also have
m
X
kσke∗j = σ(e∗j ) =
γi (e∗j )e∗i
Pm
j=1
γj (h)e∗j . We
i=1
and hence
Now
γi (e∗j )
= kσkδij .
kγi k = sup |γi (h)| = sup |σ(h)(ei )| ≤ sup kσ(h)k = kσk
khk≤1
=
m
X
khk≤1
kσkδij =
j=1
m
X
khk≤1
|γi (e∗j )| ≤ kγi k
j=1
and thus
kγi k =
m
X
|γi (e∗j )|.
j=1
Therefore, by Theorem 3.2
Z
γi (h) = kσk
h(x, ei ) dµi
X
for some Borel probability measure µi on X.
∗
Theorem 4.2. Let γ ∈ C . Then kγ | Lk = kγk if and only if there exists
σ ∈ C # and w ∈ Rm such that
σ(T ) = kσkT
for all T ∈ L and
γ(h) = σ(h)(w)
for all h ∈ C.
Proof. Suppose kγ | Lk = kγk. Then by Theorem 3.2
γ(h) =
m
X
j=1
γ(e∗j )
Z
X
h(x, ej ) dµj
132
H. B. KEYNES, N. G. MARKLEY and M. SEARS
where µ1 , · · · , µm are Borel probability measures on X. Define σ ∈ C # by
σ(h) =
m Z
X
j=1
X
h(x, ej ) dµj e∗j ,
set w = (γ(e∗1 ), · · · , γ(e∗m )) and check the details.
For the converse observe that
γ(e∗j ) = σ(e∗j )(w) = kσkwj
and
kγk = sup |σ(h)(w)| ≤ kσk |w| =
khk≤1
m
X
|γ(e∗j )| = kγ | Lk ≤ kγk.
j=1
This completes the proof.
Definition 4.3. An invariant cocycle integral is an element σ of C # satisfying:
1) kσk = 1
2) σ(ah) = σ(h) for all h ∈ C and a ∈ Zm
3) σ(T ) = T for all T ∈ L.
The set of invariant cocycle integrals for (X, Zm ) will be denoted by I(X, Zm ) or
simply I.
Theorem 4.4. Let σ ∈ C # . Then σ ∈ I if and only if there exist invariant
Borel probability measures µ1 , · · · , µm such that
σ(h) =
m Z
X
j=1
X
h(x, ej ) dµj e∗j .
Proof. Again define γi (h) = σ(h)(ei ). Note that γi (ah) = σ(ah)(ei ) = σ(h)(ei )
= γi (h) for all a and h. Consequently by Theorems 3.4 and 4.1 it has the required
form. The converse is just a matter of checking that properties 1), 2) and 3) hold.
We do not know whether or not the measures µj are uniquely determined by σ.
In other words are there always enough cocycles so that the functions h(·, ej ) can
distinguish invariant measures?
It is also now a routine calculation to prove the following:
Theorem 4.5. If w ∈ Rm with |w| = 1, then
I(w) = {γ : ∃ σ ∈ I 3 γ(h) = σ(h)(w)}.
ERGODIC AVERAGES AND INTEGRALS OF COCYCLES
133
Theorem 4.6. The set I of invariant cocycle integrals is a non-empty convex
weak∗ compact subset of C # . Moreover, there is an affine weak∗ homeomorphism
Φ from I onto I(e1 ) × · · · × I(em ).
The ideas and formulas in Sections 3 and 4 can also be expressed in the language
of tensor products. In particular, C # is naturally identified with the continuous
bilinear forms on C × Rm and hence is isomorphic to (C ⊗ Rm )∗ . Working from
this point of view K. Madden [6] has shown that C # is naturally isometrically
isomorphic to the Banach space of all bounded linear functions ρ : Cm → Lm such
that ρ(T g) = T ρ(g) for g ∈ Cm , the Rm valued cocycles, and T ∈ Lm , the linear
maps of Rm into itself.
5. Extreme Points and Ergodic Averages
Let µ ∈ M, the invariant Borel probability measures for (X, Zm ). So M is a
compact convex set in the weak∗ topology and its extreme points are the ergodic
measures E. If µ ∈ E, then by the ergodic theorem there exists x ∈ X such that
for all f ∈ C(X)
Z
X
1
f dµ = lim
f (ax)
m
N
→∞
N
X
0≤ai <N
1≤i≤m
and the ergodic averages provide an access to the extreme points of M and hence
to M. In this section we establish two similar approaches to the extreme points
of I(ej ).
For N ∈ N, x ∈ X, and h ∈ C let AN σx denotes the element of C # defined by
AN σx (h) =
m
1 X
N m j=1
X
h(ax, ej )e∗j .
0≤ai <N
1≤i≤m
It is easily checked that kAN σx k = 1, and AN σx (T ) = T for T ∈ L. If ANk σxk
converges to σ in the weak∗ topology and Nk goes to infinity, then σ ∈ I because
|AN σx (ek h) − AN σx (h)| ≤
2mN m−1
2mkhk
khk =
.
m
N
N
Let σ be a weak∗ limit of a sequence ANk σxk with Nk going to infinity. ANk σxk
can be regarded as acting on C(X) so that σ ∈ C ∗ (X). Thus there exists µ ∈ M
such that
m Z
X
σ(h) =
h(x, ej ) dµe∗j .
j=1
X
The map Θ : C → C(X)m has a dual Θ∗ : C ∗ (X)m → C ∗ defined by Θ∗ (γ)(h) =
γ(Θ(h)) and we can look at this coordinate-wise, that is θj∗ : C ∗ (X) → C ∗ by
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H. B. KEYNES, N. G. MARKLEY and M. SEARS
θj∗ (η) = Θ∗ (0, · · · , η, · · · , 0) or θj∗ (η)(h) = η(h( · , ej )). It follows from Theorem 3.4
∗
Rthat θj (M) = I(ej ), and the extreme points of I(ej ) all have the form γ(h) =
X h(x, ej ) dµ for some µ ∈ M. (In fact, we will see in a minute that we could
even assume µ is ergodic.)
Because there is an affine homeomorphism from I onto I(e1 ) × · · · × I(em ), the
extreme points of I have the same structure as the extreme points of a cartesian
product of compact convex sets. As soon as two of the sets I(ej ) contain more
than one point, there exists an extreme point σ of I that does not have the form
of σ(h) shown above with a single µ ∈ M. Consequently we turn our attention to
the extreme points of I(ej ).
Theorem 5.1. If γ is an extreme point of I(ej ), then there exists x ∈ X such
that AN σx converges in the weak∗ topology to σ ∈ I satisfying γ(h) = σ(h)(ej ).
Proof. Consider the compact convex set {µ ∈ M : θj∗ (µ) = γ}. It is easily
checked that its extreme points are also extreme point of M. Hence there exists
µ ∈ E such that θj∗ (µ) = γ and the required σ is given by
σ(h) =
m Z
X
j=1
X
h(x, ej ) dµe∗j .
For m = 1 and a = ne1 , n > 0 we have
n−1
h(x, a)
1X
h((ke1 )x, e1 )
=
|a|
n
k=0
which is an ergodic average. How do we interpret h(x, a)/|a| as an ergodic average
in general? We will show that when a goes to infinity in a particular direction,
h(x, a)/|a| will determine an invariant integral in that direction.
Given x ∈ X and a ∈ Zm , a 6= 0, define γ(x,a) ∈ C ∗ by
γ(x,a) (h) =
h(x, a)
.
|a|
Pm
Obviously, kγ(x,a) k ≤ 1 because |h(x, a)| ≤ khk |a|. Using T = j=1 sign(aj )e∗j
∈ L, we have kT k = 1 and γ(x,a) (T ) = 1. Hence kγ(x,a) k = kγ(x,a) | Lk = 1.
Theorem 5.2. Let {xk } and {ak } be sequences in X and Zm . If γ(xk ,ak )
converges to γ in the weak∗ topology and |ak | goes to infinity, then ak /|ak | converges
to w and γ ∈ I(w).
Proof. Observe that
(ak )j
,
k→∞ |ak |
γ(e∗j ) = lim γ(xk ,ak ) (e∗j ) = lim
k→∞
ERGODIC AVERAGES AND INTEGRALS OF COCYCLES
135
and hence limk→∞ ak /|ak | converges to w = (γ(e∗1 ), · · · , γ(e∗m )). Clearly |w| = 1
Pm
and kγk ≤ 1 = j=1 |γ(e∗j )| = kγ | Lk ≤ kγk. So kγk = 1 and it remains to show
that γ is invariant.
It suffices to show that γ(ej h) = γ(h), which is equivalent to showing that
h(ej xk , ak ) − h(xk , ak )
0 = lim γ(xk ,ak ) (ej h) − γ(xk ,ak ) (h) = lim
.
k→∞
k→∞
|ak |
This follows from
|h(ej xk , ak ) − h(xk , ak )| = |h(xk , ak + ej ) − h(xk , ej ) − h(xk , ak )|
≤ |h(xk , ak + ej ) − h(xk , ak )| + |h(xk , ej )|
≤ khk |ak + ej − ak | + khk |ej | = 2khk
because |ak | goes to infinity.
Let Mj denote the invariant Borel measures for Tj (x) = ej x and let Ej denote
Tm
the ergodic measures for Tj . So M = j=1 Mj .
Theorem 5.3. For j = 1, · · · , m, I(ej ) = θj∗ (Mj ). If γ is an extreme point of
I(ej ), there exists x ∈ X such that γ(x,N ej ) converges to γ in the weak∗ topology.
Proof. Because M ⊂ Mj we know I(ej ) ⊂ θj∗ (Mj ). Since θj∗ is linear, it
suffices to show that θj∗ (Ej ) ⊂ I(ej ) to prove that θj∗ (Mj ) = I(ej ).
If µ ∈ Ej , then because µ is ergodic there are points x ∈ X such that for all
f ∈ C(X)
N −1
1 X
µ(f ) = lim
f (Tjk (x)).
N →∞ N
k=0
Thus for h ∈ C
θj∗ (µ)(h) = lim
N →∞
h(x, N ej )
= lim γ(x,N ej ) (h)
N →∞
N
and by the previous proposition the weak∗ limit of γ(x,N ej ) is in I(ej ).
To prove the second part proceed as in the proof of Theorem 5.1.
6. Uniform Convergence of Ergodic Averages
In this section we will show that for uniform convergence the two types of
ergodic averages, AN σx (h) and h(x, a)/|a|, have the same behavior. The invariant
cocycle integrals provide the link between them and clean generalizations of the
classical results for Z. These results then lead to characterizations of a unique
cocycle integral and of the closure of the coboundaries.
136
H. B. KEYNES, N. G. MARKLEY and M. SEARS
Proposition 6.1. Let h ∈ C. The ergodic average
1
Nm
X
h(ax, ej )
0≤ai <N
converges to 0 uniformly in x if and only if γ(h) = 0 for all γ ∈ I(ej ).
Proof. Suppose it does not converge uniformly to 0. Then there exists > 0
and sequences {xk } and {Nk } in X and Z such that Nk goes to infinity and
|ANk σxk (h)(ej )| ≥ .
We can assume without loss of generality that ANk σxk converges in the weak∗
topology to σ ∈ I. Then clearly |σ(h)(ej )| ≥ and γ(h) 6= 0 for all γ ∈ I(ej ).
Now suppose the average does converge uniformly to 0. It suffices to show
that γ(h) = 0 for every extreme point of I(ej ). If γ is an extreme point of
I(ej ), then using Theorem 5.1 it follows from the uniform convergence to 0 that
γ(h) = σ(h)(ej ) = 0.
Proposition 6.2. Let h ∈ C and let w ∈ Rm with |w| = 1. If γ(h) = 0
for all γ ∈ I(w), then h(x, ak )/|ak | converges to 0 uniformly in x whenever
limk→∞ ak /|ak | = w and limk→∞ |ak | = ∞.
Proof. Suppose the convergence is not uniform to 0. By choosing a subsequence
of ak we can assume
|h(xk , ak )|
≥
|γ(xk ,ak ) (h)| =
|ak |
for some > 0 and a sequence {xk } from X. Then by taking a weak∗ limit of a
subsequence of γ(xk ,ak ) , we obtain γ ∈ I(w) such that |γ(h)| ≥ .
The converse holds for w = ej .
Proposition 6.3. Let h ∈ C. The sequence h(x, nej )/n converges to 0 uniformly in x as n goes to infinity if and only if γ(h) = 0 for all γ ∈ I(ej ).
Proof. The “if” part follows from the previous proposition. Assume the convergence is uniform and let γ be an extreme point of I(ej ). Then by Theorem 5.3
γ is the weak∗ limit of γ(x,N ej ) with N going to infinity, and
h(x, N ej )
= 0.
N →∞
N
γ(h) = lim
Propositions 6.1 and 6.3 have the following corollary
ERGODIC AVERAGES AND INTEGRALS OF COCYCLES
137
Corollary 6.4. Let h ∈ C. The ergodic average
1
Nm
X
h(ax, ej )
0≤ai <N
converges to 0 uniformly in x as N goes to infinity if and only if the sequence
h(x, nej )/n converges to 0 uniformly in x as n goes to infinity.
These three propositions can also be assembled into the following global result.
Theorem 6.5. Let h ∈ C. The following are equivalent:
(1) The average AN σx (h) converges to 0 uniformly in x as N goes to infinity.
(2) As |a| goes to infinity h(x, a)/|a| converges to 0 uniformly in x.
(3) σ(h) = 0 for all σ ∈ I.
The now obvious proof that (3) implies (2) is precisely where a real advantage
has been gained from the use of I. The next theorem is an application that builds
on the full strength of Theorem 6.5.
Theorem 6.6. The following are equivalent:
(1) There exists a unique invariant cocycle integral σ0 .
(2) For h ∈ C there exists Th ∈ L such that AN σx (h) converges uniformly in
x to Th as N goes to infinity.
(3) For h ∈ C there exists Th ∈ L such that
|h(x, a) − Th (a)|
|a|
goes to 0 uniformly in x as |a| goes to infinity.
Proof. Assuming (1) holds we can apply Theorem 6.5 to h − σ0 (h) because
σ0 (h−σ0 (h)) = σ0 (h)−σ0 (h) = 0 and obtain AN σx (h−σ0 (h)) = AN σx (h)−σ0 (h)
converging uniformly to 0. Similarly (2) implies (3) because Theorem 6.5 can be
applied to h − Th. Finally if (3) holds, Theorem 6.5 can be used again to conclude
that σ(h − Th ) = 0 for all σ ∈ I. It follows that σ(h) = Th for all σ ∈ I and hence
I contains at most one element.
If σ ∈ I is given by a single ergodic measure, that is
m Z
X
σ(h) =
h(x, ej ) dµe∗j
j=1
X
where µ is an extreme point of M, then the results of Boivin and Derriennic [1]
apply. In particular, their Theorem 1 implies that for h ∈ C
|h(x, a) − σ(h)(a)|
=0
|a|
|a|→∞
lim
138
H. B. KEYNES, N. G. MARKLEY and M. SEARS
µ-almost everywhere. Theorem 5.3 can be interpreted as a generalization of this
result, but may not be the best possible. Although working with elements of I
given by several invariant measures may seem cumbersome, these cocycle invariant
integrals do lead to a characterization of B as the next theorem shows.
Our final result is to use I to identify the subspace B of C. As in Section 2,
define S : C(X) → C by Sf (x, a) = f (ax) − f (x) and B is the set of coboundaries
of the range of S. Clearly σ(Sf ) = 0 and hence σ(h) = 0 for all h ∈ B. We will
prove the converse.
The following calculation which H. Furstenberg showed us will be essential.
Given h ∈ C, set
X
1
fN (x) = − m
h(x, a).
N
0≤ai <N
Clearly fN ∈ C(X). Now set
hN = h − SfN
so hN and h differ by a coboundary.
Now using the cocycle formula we have
X
1
hN (x, ej ) = h(x, ej ) + m
{h(ej x, a) − h(x, a)}
N
0≤ai <N
1
= h(x, ej ) + m
N
X
{h(ax, ej ) − h(x, ej )}
0≤ai <N
= AN σx (h)(ej ).
We can now prove:
Theorem 6.7. B = {h : σ(h) = 0 for all σ ∈ I}.
Proof. We have already pointed out that σ(h) = 0 for all σ ∈ I when h ∈ B.
Suppose that σ(h) = 0 for all σ ∈ I. By Proposition 6.1 AN σx (h)(ej ) converges
to 0 uniformly in x for j = 1, · · · , m. But hN (x, ej ) = AN σx (h)(ej ), so
kh − SfN k = khN k
converges to 0 and h ∈ B.
Theorem 6.8. There is a unique invariant cocycle integral if and only if C =
L ⊕ B.
Proof. If I = {σ0 }, then Theorem 6.7 applies to h − σ0 (h). If C = L ⊕ B, then
h ∈ C can be uniquely expressed as h = T + g with T ∈ L and g ∈ B. Hence
σ(h) = T for σ ∈ I and there is a unique cocycle integral.
Clearly unique ergodicity implies there is a unique cocycle integral. The converse is an interesting open question.
ERGODIC AVERAGES AND INTEGRALS OF COCYCLES
139
References
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Theory and Dynamical Systems, (to appear).
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and growth properties of Zn cocycles, Proceedings of the London Math Society (3) 66 (1993).
3. Hedlund G., A class of transformations of the plane, Proceedings of the Cambridge Philosophical Society 51 (1955), 554–564.
4. Keynes H., Markley N. and Sears M., On the structure of minimal Rn actions, Quaestiones
Mathematicae 16 (1993), 81–102.
, On close to linear cocycles, Proceedings AMS, (to appear).
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6. Madden K., The Existence and Consequences of Exotic Cocycles, Thesis, University of Maryland, 1994.
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9. Walters P., An introduction to ergodic theory, Graduate texts in Mathematics 79, SpringerVerlag.
H. B. Keynes, School of Mathematics, University of Minnesota, Minneapolis, MN 55455, U.S.A.
N. G. Markley, Department of Mathematics, University of Maryland, College Park, MD 20742,
U.S.A.
M. Sears, Department of Mathematics, University of the Witwatersrand, Johannesburg, South
Africa